Using Feynman's technique TWICE! (the integral of sin^3(x)/x^3 from 0 to inf)

We will evaluate the improper integral of sin^3(x)/x^3 from 0 to infinity by using Feynman's technique of integration (aka differentiation under the integral sign, Feynman's integration trick, or Leibniz's Rule). This is definitely one of the coolest integration techniques (but unfortunately it is not often taught in a calculus class).

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[lordofhunger]

We need a tutorial about where to use each pen

[weinihao]

This kind of video, where you show your thought process and consider which route to go and even hit a dead end is very very nice as it teaches how to tackle the problem instead of simply presenting a deus ex machina solution.

[LuigiElettrico]

Looking at the clock and hearing it being synchronized with my own wall clock makes me feel like I am in the class :D Great integral!

[omograbi]

10:14 shouldn't it be sin(3tx)/x?
Or it's anyway the same answer?

[vatsalmalav]

I like how you say "this guy" making numbers look like living things that make your life easier and many times Hard. This is a great integral you solved I loved it.

[ritvikg]

10:16 I didn't get this step. Firstly how did he replaced the sin(3tx) with just sin(tx) and in the next step after substituting tx as u, he should get a 't' after integration which he missed as well. It should have been -3πt/8 + 9πt/8 considering his previous step of omitting 3 from the sin is right. Can anyone help me with this.

[yutaj]

Expressing sin³(𝑥) in terms of sin(3𝑥) and sin(𝑥) using the triple angle formula in the first place seems helps.

[jamiewalker]

The integral - after using the fact that the integrand is even, using the triple angle formula can be written as 1/8 Im{ integral (e^i3x - 3e^ix)/x^3 dx } where the integral runs from -infinty to infinity. We can analytically continue into the complex plane, separate the two integrals, run a contour along an infinite semi-circle in upper half plane, and a small-semi circle in upper half plate, circulating the singlarities at z = 0 of the function. Using Jordan's lemma to determine that the integral around the large semi-circle is 0, and using Cauchy residue (no poles within contour) means that the integral is equivalent to integrating in the complex plane the above integral around an infinitely small semi-circle, centred at z = 0. The result is 1/8 Im(0.5*2*i*pi*residue at z = 0). The residue, of the above integrand at z = 0 is -3 (which can be quickly checked by expanding the exponential numerator to quadratic term. Plugging this in gives the answer...no Feynman...

[drpeyam]

Reminds me of the Borwein integrals a bit

[zunaidparker]

If you Laplace transform this integral you'll see why the value for the 3rd power is different from the first two powers. Essentially you're doing a convolution, which amounts to taking a moving average over a sliding window of a rectangular function. For the first 2 powers, the window isn't wide enough to affect the value of the average over the moving window, but for the 3rd power, eventually we are averaging zero contributions from outside the rectangle which brings the moving average down.

[hrv]

14:30 man i know that happiness and you have to experience it atleast once in a lifetime!

[-fai]

Hey sir, Feynman's technique is mad cool but... Where should I set the parameter? Is there any "rule" to follow?
I mean, you are supposed to put the parameter on a place in which after deriving, the integral is easier to solve, but it would be marvelous if you have a structured guide that tells you where to put it depending on the situation. It would be great a video like... "When and where to use Feynman's technique"
Thanks sir.

[mikefochtman]

I lost something at 10:16. You got 9/4 int(0, inf) sin(tx)/x. But the previous step it was 3/4 sin(3tx)3x/x^2 ?? Shouldn't the argument to the sin function still be 3tx? How did that 3 disappear?

[sngash]

This is excellent and the video led me to Math 505's generalized version of sin^n(x)/x^n which looks like a great beast for you to work your magic on and possibly make understandable at around calc 2 level :). I struggled following the differentiation portion

[sikf]

Everytime I watch one of your videos, it feels like an emotional rollercoaster.

[wcottee]

Maybe I missed it but at 10:16 how did the second term go from sin(3tx) to just sin(tx)?

[vogelvogeltje]

You have 60hz hum coming from your microphone JSYK. Try to turn your microphone up more without clipping over 0dbFS, or look and see if any wires are crossing over a power wire from your interface.

[boomgmr]

At 11:54 you dont write sin3tx again, is that a mistake?

[AntimatterBeam]

I just bought calculus clothing off your store. My weird maths science wardrobe is increasing. I am happier than I was 30 mins ago now.

[abdulmalek]

Hello ! I hope you see my comment
I saw this nice question so that I recommend it
The question is : solve the system of equations
a = exp (a) . cos (b)
b = exp (a) . sin (b)
It can be nicely solved by using Lambert W function after letting z = a + ib
Hope you the best ... your loyal fan from Syria